Pop. g. stats2.doc -
9/3/05
Shane’s Simple Guide to F-statistics
Intro
The aim here is simple & very focussed:
Aim : a very brief introduction to the most common statistical methods of analysis of population genetic
structure (i.e., F-statistics and AMOVA), and how to interpret them.
Disclaimer!
Please remember that the following discussion is very simplified, describes only one of many approaches
to F-stats, and ignores many assumptions and interrelated material. To ease understanding, some
explanations are not 100% accurate, but should not be misleading. You will need to check text or original
sources (or ask me) for further details and qualifications.
Background material
Terms
You need to familiar with what is meant by the following terms:
•
•
•
•
heterozygosity
homozygosity
inbreeding – what it means in terms of genotype frequencies
effective population size - Ne
Random genetic drift
This is a central concept to understand in the genetic divergence of populations. To refresh your memory,
one simple way of thinking of how genetic drift occurs from one generation to the next is shown below.
(Note – where generations do not overlap, Ne can be thought of as the effective no. of breeders.) When
only a small number of breeders contribute to the next generation, the small number of randomly-selected
genes that are passed on are likely to differ slightly in their frequencies from the previous generation,
simply due to the random sampling process.
Change of gene frequency
over one generation
(Genetic Drift)
Freq. of
black allele
1st
generation
Indivs that
successfully mate
(N e)
2nd
generation
0.25
0.20
0.20
2
The degree of genetic drift from one generation to the next depends on how large are the number of
breeders (Ne). The smaller Ne , the larger the drift in frequencies from one generation to the next is likely
to be:
Effect of No. of breeders (Ne)
on genetic drift
Large no. of
breeders
1st
generation
Indivs that
successfully mate
(N e)
2nd
generation
Small no. of
breeders
Freq.
Freq.
0.25
0.25
0.20
0.10
0.20
0.10
The cumulative effect of this genetic drift over many generations is shown in the figure below. Each line
shows the drift in frequency over time of one allele in one subpopulation. It can be seen that (with no
migration) the populations gradually diverge genetically more and more over time. This effect is greater,
obviously, for populations with smaller Ne.
3
Hardy-Weinberg Equilibrium
This is also a central concept in the derivation of F-statistics. The most basic point here (as illustrated
below) is that, regardless of the genotype frequencies you start with in one generation, if there is
completely random mating, then the genotype frequencies of the next generation will tend towards a
highly predicted ratio - the HWE – that is determined entirely by the allele frequencies.
7AA
+
3BB -> 14A & 6B
freq. [A] = p = 0.7
freq. [B] = q = 0.3
OR
5AA + 4AB + 1BB -> 14A & 6B
A
A
A
A
B
A
A
B
A
A
B
A
A
A
A
A
B
A
A
B
A
B
A
A
A
B
B
A
A
B
A
A
B
A
B
A
B
A
A
A
5AA + 4AB + 1BB
But how is this genotype ratio derived? Very simply, it comes from the probabilities of getting each of the
three types of allele pairs (genotypes) shown above. Each of these three combined probabilities comes
merely from the product of the probabilities (or frequencies) of the two alleles. This is shown below in
both example numbers and in terms of the generalised allele frequencies, p and q. You might recognise
that the formulae for the genotype frequencies are in the form of a binomial expansion. Thus, with more
than two alleles the H-W frequencies can be easily determined using the appropriate expansion.
It can also be seen below that the formulae for calculating expected (as opposed to observed)
heterozygosity and homozygosity come straight from the HWE. It is useful to note here that when there
are more than 2 alleles, it is easier to calculate heterozygosity (H) using H = 1 - homozygosity
H-W calculation
4
p = 0.7 q = 0.3
A
2
B
0.7
(p)
A
.7
(p2)
.7x.3
(p.q)
0.3
(q)
B
.7x.3
(p.q)
.32
(q2)
genotypes :
AA
AB
frequency:
0.49 +
2 x 0.21
p2
BB
+
0.09 = 1.0
2p.q
q2
(binomial expansion)
= 1.0
Homozygosity = p2 + q2
Heterozygosity = 2p.q
or = 1 - homozygosity
= 1 – (p2 + q2)
or, if > 2 alleles = 1 – (p12 + p22 + p32 + … pn2)
= 1 - ! pi2
Effects of population sub-division on heterozygosity
One of the main effects that population subdivision has on genetic diversity, is the reduction in observed
H compared with expected H. This is shown in the following two examples:
Examples
• Mice (Hartl 1997 p. 112):
5
•
Blue & white flowers (Hartl 1997 p. 114)
Derivation and explanation of F-statistics
The extent of reduction in observed H shown in these examples above can be used to quantify the level of
genetic differentiation between the subpopulations. This quantification has been formalised (in the first
instance by Wright (1951, 1965, 1978) in a series of hierarchical F-statistics.
6
Firstly, a series of hierarchical measures of heterozygosity are defined:
HI = mean observed heterozygosity per individual
within subpopulations
HS = mean expected heterozygosity within random mating subpopulations = 2piqi
HT = expected heterozygosity in random mating total population = 2 p q
The quantities that these terms are measuring can be visualised as below:
___
HI = HO
HS = 2 pi qi
__
HT = 2 p q
Now, using these three different hierarchical measures of H, we can define three hierarchical F-statistics,
defined below:
•
INBREEDING COEFFICIENT = FIS = (HS - HI) / HS
- the mean reduction in H of an individual due to non-random mating within a
subpopulation
- i.e., a measure of the extent of genetic inbreeding within subpopulations
- can range from –1.0 (all individuals heterozygous) to +1.0 (no observed heterozygotes)
- sometimes referred to simply as F rather than FIS
•
FIXATION INDEX = FST = (HT - HS) / HT
- the mean reduction in H of a subpopulation (relative to the total population) due to
genetic drift among subpopulations
- i.e., a measure of the extent of genetic differentiation among subpopulations
- can range from 0.0 (no differentiation) to 1.0 (complete differentiation – subpopulations
fixed for different alleles)
7
•
OVERALL FIXATION INDEX = FIT = (HT - HI)/HT
- the mean reduction in H of an individual relative to the total population
Note: FIT combines contributions from non-random mating within demes (FIS) and
effects of random drift among demes (FST) –
The relationship between the three F-statistics is:
(1 - FIT) = (1 - FIS) (1 - FST)
Now, going back to our two initial examples in Hartl, we can see what FST means in a couple of real,
simplified examples.
Mouse example:
FST =
=
HT ! HS
HT
(H T = 2pq = 2 " .5 " .5 = 0.5)
0.5 ! 0.0
= 1.0
0.5
that is, there is absolute differentiation between the 2 subpopulations., with each fixed for a
different allele. Another way of thinking of this (obvious from fig 4.1) is that 100% of the total genetic
variation is between subpopulations., with zero variation within subpopulations
Flower example (all subpopulations – not considering regions):
FST =
0.2371 ! 0.1424
= 0.39
0.2371
That is, there is a substantial differentiation among all the subpopulations as can be seen from the
great variation in allele frequencies in fig. 4.2. Putting this in other words, 39% of the total genetic
variation is distributed among subpopulations, with 61% of the variation within subpopulations
Although FST has a theoretical range of 0 to 1.0, the observed maximum is usually much less than 1.0.
(See below for the effect of highly variable loci such as microsatellites.) Wright (1978) suggests the
following qualitative guidelines for the interpretation of FST (based on allozyme loci):
- the range 0.0 to 0.05 may be considered as indicating little genetic differentiation
- the range 0.05 to 0.15 indicates moderate genetic differentiation
- the range 0.15 to 0.25 indicates great genetic differentiation
- values of FST above 0.25 indicate very great genetic differentiation
8
However, keep in mind that these are very general guidelines.
Table 3 (p.302) from Hartl (1989) gives some comparisons among FST values from a range of species to
give some perspective as to what can be expected in natural populations.
Extension to hierarchical F-statistics
It can be seen conceptually, without too much difficulty, that these three F-statistics described above
could be extended to include higher levels of hierarchy. For example, if we have a series of
subpopulations which naturally fall into three separate groups (e.g., 3 river or ocean basins), we could add
another level, groups, in our hierarchy to give: (1) variation among indivs. within subpopulations, (2)
subpopulations, (3) groups of subpopulations, and (4) total variation. If we denote the group with
subscript C (as done in Arlequin), this would give us the following FST –related statistics:
FST – the variance among subpopulations relative to the total variance
FSC – the variance among subpopulations within groups
FCT – the variance among groups relative to the total variance
The calculation of these values is a relatively straightforward extension of those shown previously. This
hierarchy could be extended upwards further, if suitable for the data set. It could also be extended
downwards, to consider variation within individuals, if you have diploid genotypic data.
Modifications to FST calculations required
The basic calculation formulae for F-statistics given above were derived as simple theoretical predictions.
However, as with all statistics, we are trying to estimate a theoretical quantity using limited and imperfect
data. That is, all we can actually do is to calculate an estimate, FST-hat, which hopefully closely
approximates the true FST . To achieve as close an approximation as possible, we need to allow for a
number of factors, listed below.
9
Multiple alleles and loci
The original formulation of FST by Wright considered only one biallelic locus. This was extended to first
accommodate multiple alleles, and then to accommodate multiple loci (Wright 1978, Nei 1973). (The
multiple locus version was termed GST by Nei, but I recommend sticking to FST for consistency). The
formulae I have presented above are the multiple-allele forms. The multiple locus forms may be
calculated in a number of ways, the simplest being to just average FST over loci, although it is more
appropriate to average the HS and HT over loci. It seems an appropriate place here to emphasise how
important it can be to estimate FST using as many loci as possible. Each locus is like a totally independent
random trial. Among a group of subpopulations, each locus may, purely by chance, drift in frequency
quite differently from the next locus. As a result, each locus will provide a different estimate of FST ,
sometimes radically so. Examples of this are shown in the tables in Box F (pp300-301, Hartl 1989). It is
clear that an average over many loci will give a much better measure of differentiation among populations
than a calculation from only one locus, such as the mt locus.
Sampling effects
There also need to be adjustments made to these statistical calculations to account for the errors
introduced by limited sampling. The first level of sampling to be considered is the limited sampling of
individuals within a subpopulation Sometimes it is also necessary to account for the limited sampling of
subpopulations from the total number of subpopulations within the species. This is necessary when you
are making inferences about differentiation among all the subpopulations (a random effects model) (e.g.,
“This study therefore implies that there exists significant population structure among all subpopulations
of species X”). This random effects model is usually used in ANOVA / AMOVA approaches (see below).
However, this sampling correction is not strictly necessary if you are making inferences about only those
populations you have sampled (a fixed effects model) (e.g., “This study therefore implies that there exists
significant population structure among subpopulations A, B & C of species X”). The specific mechanisms
of incorporating sampling error into your calculations won’t be dealt with here, because they become
quite complicated, but it is important that you know whether the program you are using to calculate FST ‘s
does incorporate such corrections (and what they are).
Haploid Data
The explanation given above using heterozygosity for determining FST for diploid loci is all well and
good, but how do you define FST for a haploid locus, where heterozygosity is relatively meaningless? This
has been approached in a number of ways (as usual), but the simplest way of reconceptualising FST for
haploid loci is to think in terms of haplotype diversity instead of heterozygosity. Haplotype diversity
(conveniently, also referred to as H) is a measure of the degree of variation in haplotypes found within a
population, and is calculated as:
j
H = 1! " pi 2
i =1
This just happens to be the same way we can calculate heterozygosity in a diploid locus, although there
are no heterozygotes here. To give you a quick conceptual idea of what haplotype diversity means in a
real example, here is a worked example of two subpopulations, where the second subpopulation is
10
intuitively less diverse in haplotypes than the first. (I use the terms ‘haplotype’ and ‘allele’
interchangeably).
Pop. 1
Haplotype
1
2
3
H
pi
.5
.4
.1
Pop. 2
2
i
p
.25
.16
.01
Σpi2=.42
1-Σpi2=0.58
pi2
.81
.01
pi
.9
.1
Σpi2=..82
1-Σpi2=0.18
So, to now calculate FST for a haploid locus, just replace the H (heterozygosity) terms with the equivalent
H (haplotype diversity) terms in the original formula above.
DNA Sequence Data
So far, we have been concerned only with allele (or haplotype) frequencies when calculating F-statistics.
This is fine for allozyme and microsatellite data, where this is all the information we have. However,
when we also have DNA sequence (or RFLP) data, we can determine how different each haplotype (or
allele) is from each other, which we know gives us a lot more information about population substructure
than we get from purely the haplotype frequencies. How can we incorporate this additional information
into a similar measure of subpopulation differentiation?
Slightly different measures have been devised by different authors (as usual!), but the simplest of these to
comprehend in light of our previous discussion is probably that of Nei (1982). What he did was to define
a similar measure of population differentiation as FST , but this time using a measure of nucleotide
diversity (π) within a population, in place of heterozygosity (H) or haplotype diversity (H). If we define
πij as the genetic distance between haplotype i and haplotype j (measured either by the simple proportion
of nucleotide differences, or by some more complicated method, e.g., Jukes-Cantor, Kimura 2-parameter,
etc.), then the nucleotide diversity within the total population is
! T = " pi p j ! ij
ij
where pI and pj are the overall frequencies of haplotypes i and j respectively. That is, the distances
between haplotype pairs are simply weighted by how common they are, to arrive at an average. If we also
define πS as the average nucleotide diversity within subpopulations, then we can derive a familiar
expression for an FST –like nucleotide measure of subpopulation differentiation:
!T " ! S ! B
=
!T
!T
11
Here, πB is the average nucleotide diversity between subpopulations What all these terms refer to
conceptually can be seen in the following figure:
π S2
π S1
_
π S=(π S1+π S2+π S3)/3
π ij
πB
πT - πS = πB
π S3
π T=
+
This statistic could also be called FST , but it was originally described by Nei (1982) as γST . A related
statistic derived by Lynch & Crease (1990) was called NST , one derived for mtDNA data by Takahata and
Palumbi (1985), GST , and one by Excoffier et al. (1992, see below), ΦST (phi-st). Although each of these
statistics for nucleotide data is calculated slightly differently, in reality they are all trying to estimate the
same parameter - the proportion of nucleotide diversity among subpopulations, relative to the total – and
their values are usually quite similar, particularly with larger sample sizes. The same things can be said
for all the different ways of calculating FST from allelic data (including GST and θST ). Given the multitude
of different descriptor variables used by different authors, and the fact that within the two classes they are
trying to estimate essentially the same parameters, my convention is to refer to the allelic form of the
statistic as FST , and the nucleotide diversity form as ΦST , and simply mention somewhere whose
formulae or program you used to calculate them.
There is also a very simple conceptual relationship between the allelic FST and the nucleotide ΦST , shown
below. In the allelic calculations (FST ), it is assumed that all alleles are equidistant from each other, while
in the nucleotide diversity calculations (ΦST ), there are different distances between different alleles. (This
can indeed be applied to any other type of distance calculation you might require, such as between
microsatellite alleles). In fact, now we can actually calculate the allelic FST in exactly the same way we
calculate ΦST (in AMOVA – see below) by simply making all the distances between alleles equal one
(i.e., replace the pairwise distance matrix by a unity matrix). This is shown below.
FST
Φ ST
Pairwise Distances (π ij)
A
A
B
C
1
1
B
1
C
A
B
C
A
B
1
2
1
C
Below, I also provide a worked example of a simple data set, using the allele relationships shown above,
showing the difference in values of FST from the allelic form of the calculation and the nucleotide form of
the calculation.
12
Alleles
A
B
C
FST :
Σpi2=
H=1-Σpi2=
Pop. 1
Freq.
Rel. Freq.
4
.8
1
.2
0
0
Pop. 2
Freq.
Rel. Freq.
0
0
1
.2
4
.8
.82+.22=.68
1-.68=.32
.82+.22=.68
1-.68=.32
.8x.2x1=.16
.2x.8x1=.16
Total
Freq.
4
2
4
Rel. Freq.
.4
.2
.4
.42+.22+.42=.36
1-.36=.64
Φ ST :
Σpipjπ ij=
FST =
=
.4x.2x1=.08
+.2x.4x1=.08
+.4x.4x2=.32
π T=0.48
HT ! HS
HT
0.64 ! 0.32 0.32
=
= 0.5
0.64
0.64
and
! ST =
=
" T # "S "B
=
"T
"T
0.48 # 0.16 0.32
=
= 0.67
0.48
0.48
As you would expect from this data set, the ΦST value is greater, but this is not always the case, and the
reverse is true of some data sets. As explained under gene flow, differences in value between the two
forms of FST do not signify any problem or inconsistency, but are actually telling you something
interesting about your data. The two forms are really measuring different properties of your data, and one
is not necessarily any ‘better’ than the other. Even in terms of simply detecting whether or not there is
significant differentiation between subpopulations , it depends entirely on your data set as to which form
of FST - allelic or nucleotide distance – is the more powerful statistically. So the bottom line is that it is
useful to calculate both types of FST for your data set.
Microsatellites and RST
The most recent adaptation of the FST statistic has been to microsatellite data. The standard allelefrequency type of FST calculation is certainly suitable for application to this data (keeping in mind the
potential limits to the maximum value of FST in highly variable loci, mentioned below). However, if
microsatellite loci evolve in accordance with a stepwise mutation model (SMM) (Valdes et al., 1993),
then microsatellite alleles that are of similar length (i.e., have similar no. of repeats) are likely to be more
closely related to each other than alleles that are very different in length (i.e., have very different no. of
repeats). That is, microsatellite data may give us good information about the relationships among alleles,
unlike allozyme data. In the previous section I showed that, by using a ΦST –type statistic, we can use the
additional information that DNA sequence data provides about the relationships among alleles, to thence
13
provide more information about the relationships among subpopulations In a very analogous way, if we
can derive an FST -type statistic that uses the additional information that microsatellite data may provide
about the relationships among alleles, then we can obtain more information about the relationships among
subpopulations Several authors have done this ( e.g., Slatkin, 1995, Goodman, 1997) (once again, each
using slightly different calculations) and called this new statistic RST (or ρ). Slatkin defined the statistic
in terms familiar to us:
RST =
ST ! SW
ST
where ST (actually termed S-bar by Slatkin) is the variance in allele size in the total population, and SW is
the variance in allele size within subpopulations. Mikalakis and Excoffier (1996) and Rousset (1996)
have shown that this term can also be similarly estimated by ΦST in an AMOVA framework (see below),
after calculating appropriate distances between alleles (based on the sum of the squared differences in
repeat length).
The verdict is still out on whether this RST statistic is actually better than FST for microsatellites (i.e., gives
a better picture of reality). My view is that all microsatellite loci do not evolve in the same way. That is,
some loci may fit the SMM, while others do not. Also, the longer the time of divergence between
subpopulations, the greater the chance that alleles of the same length have evolved from different
ancestors (i.e., are not identical-by-descent, or, are homoplasious), or have not evolved stepwise, and thus
an SMM model may be misleading. In short, FST makes less assumptions, and thus is a more conservative
measure. You can always calculate RST as well. If this gives you very similar results, then there is no
problem. If it gives you different results, then this raises various hypotheses about the evolution of your
microsatellite loci, which you may need to look into in more detail. For example, it may be obvious from
the distributions of allele size that some or all your loci do not fit the SMM.
Alternative Derivations
The descriptions and derivations of the various F-statistics presented so far have come from just one of
the many ways in which FST can be conceptualised. Various authors have derived FST in terms of:
•
The correlation between uniting gametes, or the variance in allele frequency (Vq) (Wright’s original
derivation, 1951). Here,
FST = Vq/pq
•
The loss of heterozygosity over time in subpopulations with no migration
FST = 1 - (1 - 1/2N)t
•
The probabilities of identity-by-descent:
FST = (f0 - f)/(1 - f),
where:
f0 = Pr. identity of two alleles from same popn.,
f = Pr. identity from total popn.
•
The average times to coalescence of subpopulations (Slatkin 1991):
FST = (t - to)/t,
where
t = average time to coalescence for total population
t0 = average time to coalescence within a population
•
An ANOVA-like approach to partitioning genetic variance into within- and among-subpopulation
components (Weir and Cockerham, 1984).
14
Apart from the last of these approaches, I won’t go into any detail of these different concepts of FST here.
It is sufficient at this point to realise that they exist, at the very least so that you aren’t too confused when
you come across them. I’ll deal in more depth here with the ANOVA approach, as it has lead to the
AMOVA approach, which is now commonly used for DNA sequence data sets.
ANOVA
It may be useful here to very quickly revise the broad concepts behind the usual ANOVA procedure, as
they are much easier to conceptualise with real numbers than with gene frequencies.
So, a quick, related example of ANOVA on real numbers: comparing the mean length of whales among
subpopulations If we have i subpopulations and j individuals in each subpopulation, then our model for
partitioning variation in length among individuals is:
xij = X + aI + bij
where
and
xij
X
aI
is the length of an individual
is the overall mean length
is the subpopulation effect
with variance
bij
is the indiv. variation effect
with variance
σ2
with variance
σa2
σb 2
We test the significance of the variance among subpopulation mean lengths by calculating the ratio of the
subpopulation variance to the total variance
F = σa2 / σ2
(Although apparently similar to FST , this more familiar F is actually quite different, and can have values
greater than 1.0, unlike FST ).
Our example data set of whale lengths:
Subpopulation 1: Indiv. A: 4m, B: 5m, C: 6m
Subpopulation 2: Indiv. D: 8m, E: 9m, F: 10m
mean = 5
mean = 9
Overall mean (X) = 7
And following our model of partitioning variance, we can see the variation among individuals’ lengths as
being composed of the overall mean plus the subpopulation and the indiv. effects:
xij
x1A
.
.
.
x2F
=
X
+
aI
+
bij
= 4
=
7
+
-2
+
-1
= 10
=
7
+
2
+
1
σ2
σa2
σb 2
- related variances
Now, if we think about allele frequency variance instead of length variance, we can use a very similar
model (derived by Weir & Cockerham 1984):
15
xij = X + aI + bij
where xij
X
aI
is the probability of an individual having allele A
is the overall probability of having allele A
is the subpopulation effect
with variance
is the indiv. variation effect
with variance
σ2
with variance
σa
2
and
bij
σb 2
Once again, we measure the size of the variance among subpopulation allele frequencies by calculating
the ratio of the subpopulation variance to the total variance
FST = σa2 / σ2
This term was called θ by Weir and Cockerham (1984) (just one more variant to keep us on our toes, I
suspect). However, in this case we cannot realistically assume any particular parametric distribution of
this statistic in order to test its significance, and usually resort to non-parametric randomisation tests (see
below).
AMOVA
In a similar way that FST was adapted to γST by incorporating a measure of nucleotide distance between
alleles, so too θ was adapted to ΦST by incorporating a measure of distance between haplotypes (Excoffier
et al. 1992). A major benefit of the ΦST approach is that it is very flexible, and any type of distance you
think is appropriate can be used - hence its extension to microsatellite data also (described above).
I won’t go into too much detail here about the AMOVA approach, except to say that this is the approach
used in the ARLEQUIN program, and it is excellently documented in its manual (which I thoroughly
recommend reading if you use the program).
Comparing FST ’s
The FST statistic has proved very valuable in many ways, despite some limitations. It is a very convenient
and common way to compare differentiation among two or more subpopulations It immediately gives a
pretty good idea of the degree of subpopulation structure in any organism, without knowing anything else
at all. It also has a series of good statistical properties. These include (1) its potential relationship with
gene flow (see below), and (2) its relatively rapid approach to equilibrium - i.e., it approaches equilibrium
a lot faster than many other measures (see fig 17 on p315 of Hartl 1989), but it can still take a long time –
on the order of 1/m generations! (where m is the migration rate).
16
Perhaps one of its best attributes is that FST can be a very simple descriptor of the degree of population
substructure, for comparison among loci or species. However, this can be problematic if the level of
variation (or heterozygosity) within subpopulations varies dramatically among loci or species. FST is a
relative rather than an absolute measure of differentiation. Therefore, if some loci (or species) have very
high levels of variation compared to others, then a simple comparison of FST ‘s would be misleading. The
extreme example of this bias occurs when dealing with microsatellite loci, which often have very high
heterozygosities, sometimes approaching 1.0. Various authors have pointed out that, for a given absolute
level of subpop differentiation, highly variable loci will, by definition, give a lower value of FST than less
variable loci. As Hedrick (1999) showed,
H ! HS
H
FST = T
= 1 ! S < 1 ! HS = homozygosity
HT
HT
In other words, although the theoretical maximum value for FST is 1.0 (when each subpop is fixed for a
different allele), the practical upper limit for FST is actually the level of (expected) homozygosity. For
many microsatellite loci this means that FST must be smaller than perhaps 0.1 or less.
Another problem comparing FST –like values among loci occurs with the mitochondrial locus. Because it
is haploid and (usually) only maternally inherited, its effective population size is 1/4 that of a nuclear
locus. Thus mtDNA would be expected (on average) to diverge between subpopulations four times as
17
quickly as a nuclear locus, and hence FST values calculated from mtDNA may be four times larger than
those from nuclear data from the same populations.
The bottom line of all this discussion is that FST values can be extremely useful for comparing loci or,
more particularly, species, but that first you have to be sure you are comparing apples with apples (or
should that be kiwi fruit with kiwi fruit?).
Calculating probability and testing significance
Firstly, we need to be clear exactly what we want to test here. Usually, we want to know if there is
significant population differentiation among our samples. Hence our null hypothesis is that FST = 0, and
our alternative is that FST > 0. This need not necessarily be the case, however, as you may possibly be
interested in comparing FST ’s between groups of subpopulations or species, and therefore your H1 in this
case may be, for example, that FST < 0.2
The simplest (and earliest) method of calculating the probability of an FST value was by using the simple
equation χ2 = 2N FST , and comparing this value to the standard χ2 distribution. (It requires some
modifications for multiple alleles and loci). However, this requires us to make a big assumption about the
distribution of FST values, which is probably not valid. A very useful non-parametric approach is to
jackknife or bootstrap over loci, which provides approximate confidence intervals, and is still employed
by some programs. However, this is pretty useless unless you have something like six or more loci, and of
course is impossible if you have only one. The most flexible approach, and which is used most commonly
today, is to randomly permute (for example) individuals among the subpopulations, calculate an FST
value, and repeat this 100-1000 times to give a ‘random’ distribution of the statistic, against which you
compare your ‘true’ statistic. The random probability of getting your FST (or greater) is then simply the
proportion of the randomised values that are equal to or greater than your ‘true’ value. Using this process
you can also test hypotheses other than FST = 0, but you need to look at the actual distribution of
randomised values yourself (you can do this in ARLEQUIN).
One last, but very important, point about calculating probabilities in the context of subpop differentiation
(or generally): When a number of tests are performed at the same time, for example in a matrix of
pairwise FST ’s between subpopulations, then the probabilities actually should be adjusted for the fact that
one of the many tests may be significant simply by chance. A Bonferroni-type adjustment should be
applied to account for this (e.g., Rice 1989). My understanding is that ARLEQUIN does not do this
automatically, so you need to account for this yourself in interpreting your results. If none of this makes
any sense to you, please come and ask me, or look at a general stats book!
What FST may tell us about population divergence and gene flow
FST has proved to be a very useful parameter in many respects, as described above. One major advantage
is the possibility that it may tell us a lot about the processes leading to divergence between
subpopulations or the maintenance of that divergence. When two subpopulations begin to diverge after,
say, a vicariant event that separates them (e.g., a mountain range uplifts), then two processes begin acting,
and in opposite directions. The first process is that, under the influence of genetic drift (see fig. above),
the subpopulations start to diverge genetically. Over time, as seen in fig. 7.11 (p 286 Hartl 1997), FST will
gradually increase, until it finally approaches 1.0, if there is no continued migration between the
subpopulations.
18
On the other hand, there is likely to be some level of continued migration between the two
subpopulations, as they diverge. This migration will tend to limit the genetic divergence between the
subpopulations Thus there are two opposing forces determining the divergence between subpopulations,
and hence, FST . Genetic drift over time will allow them to diverge, while migration acts to keep them
similar (see fig below).
Subdivision of
ancestral
population
Time
( t generations)
increasing
divergence
due to drift
migration limits divergence
When we determine the level of genetic divergence between two subpopulations at a point in time, i.e.,
FST , we know that the value is due to one or both of these two effects, i.e., drift over time, or migration.
If we can assume that divergence is due entirely to one or the other effect, then we can use the value of
FST to estimate either 1) time since subpop divergence, or 2) migration. However, we cannot estimate
both. Either, 1) we assume that there has been no migration between subpopulations since divergence, in
order to estimate time since subpop divergence, or 2) we assume that divergence and FST have reached
equilibrium, in order to estimate migration. If we can make neither assumption, then we simply know that
19
the present level of divergence (FST ) is due to some combination of both effects, and therefore we can’t
estimate either parameter.
Considering the cases where we can make one of these assumptions, we will now look at the implications
in more detail.
FST and time since divergence
Given our first scenario above (that a single population splits into two subpopulations at some point,
which then each diverge randomly over time with no migration), then it is relatively easy (!) to come up
with an equation showing how FST increases over time. (For simplicity, here we will refer to FST as
simply F, and use the subscript t to indicate time in generations, i.e., Ft is the value of FST after t
generations.). This equation is
"
1 %t
Ft = 1 ! $ 1!
'
# 2N &
Fig. 7.11 (p 286 Hartl 1997) shows this relationship for various values of N. (It can be seen that for low
values of N, F rapidly approaches the limit of 1.0. In general, F goes halfway to equilibrium in 1.39N
generations). Therefore, if we know F, and N (actually Ne) then we can calculate t (in generations).
Sounds lovely doesn’t it? The only problems are that we usually have only fairly rough estimates of F and
Ne from our data, and that we must assume that our subpopulations are behaving in an ideal manner, let
alone our initial assumption of no migration!
However, all is not lost. We may have only a very rough estimate of the absolute time since divergence
for two subpopulations, but it does give us a better way to estimate relative divergence between a series
of pairs of subpopulations We see from fig 7.11 that the relationship between F and t is very non-linear. If
we transform F to make it much more linear, then we will have a much better measure of divergence
between a pair of subpopulations This is what has been done in deriving Reynolds’ distance and Slatkin’s
linearised FST (calculated in ARLEQUIN). If all the assumptions are correct, then both of these give values
in t/N generations, and are proportional to the divergence time. Such distances may be valuable for
constructing trees or other graphical patterns (such as multidimensional scaling) of the genetic
relationships among subpopulations (Other useful measures here are Nei’s total nucleotide diversity
between subpopulations (DXY) and net nucleotide diversity (DA) between subpopulations)
FST and gene flow
Now for our second scenario from above: that a single population splits into subpopulations at some
point, which then each diverge randomly over time, but in this case migration has limited the extent of
divergence, and the subpopulations have reached this limit and are at equilibrium. Now, given these
assumptions, along with the assumption that all subpopulations can share migrants with all other
subpopulations with equal chance (i.e., an island model of population structure), then there is an
extraordinarily simple relationship between FST and migration (m), which at its simplest is expressed as:
1
FST =
4Nm + 1
(The relationship between FST and Nm is shown in fig.15, p 312 Hartl 1989). The extraordinary thing
here is that the terms have simplified so that N and m appear together in one term Nm (given more
assumptions, such as m is very small, and the mutation rate, µ, is much smaller than m). This is very
satisfying (for a nerd), because 1) we no longer have to worry about trying to estimate that ever-tricky
parameter Ne ,and 2) the term Nm is actually something meaningful. Think about it – the product of the
effective population size (Ne ) and the proportion of the population that migrates (m) is the actual number
of individuals that migrate (per generation). Pretty wild isn’t it? Everyone else thinks so too, which is
why this value Nm (or just M, as referred to by Slatkin, and ARLEQUIN) has been so used and abused over
20
the years. The problem is of course that, given all the assumptions, it is pretty courageous to interpret Nm
as the true number of individuals migrating. However, as before, it is a very valuable relative measure of
migration between subpopulations Furthermore, it can also be compared with alternate estimates of Nm
that have been derived independently from the data (e.g., from coalescence approaches). All in all, it is a
pretty useful relationship, but is ultimately just a different way of expressing the same information (FST ).
FST and Φ ST over time: how allele frequency differences and nucleotide diversity differences are
not the same
To finish this brief discussion of ‘Fun with FST ’s’, I think it’s worthwhile to highlight a potentially useful
conceptual difference between the FST and ΦST parameters. Briefly, it relies on the simple concept that
allele frequencies can change quite rapidly (over only a few generations, if Ne is small), while complete
fixation of alleles takes longer, and for new alleles to arise through mutation probably longer again (when
mutation rates are relatively low). This simple series of events, therefore, is likely to take place in this
order in subpopulations, as shown diagramatically below.
What this ultimately can mean is that, after a population splits, until subpopulations have reached a stable
equilibrium (& for many organisms this is likely to not be the case at present), then FST is likely to
increase first, and only after new alleles have arisen, and monophyletic clades of alleles have begun to
arise in different subpopulations, will ΦST begin to increase substantially. That is, FST may be an
indicator of short-term or recent population processes, while ΦST may be an indicator of longer-term or
older processes.
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